| Summary: | We will analyze the symmetric positive solutions to the two-point steady state reaction-diffusion equation:
\begin{equation*}
\begin{split}
-&u^{\prime \prime}=
\begin{cases}
\lambda\left[ u-\dfrac{1}{K}u^2-\dfrac{cu^2}{1+u^2}\right];& x\in [L,1-L] ,\\
\lambda \left[u-\dfrac{1}{K}u^2\right];& x\in (0,L)\cup(1-L,1),
\end{cases}
\\
-&u^{\prime}(0) + \sqrt{\lambda}\gamma u(0) = 0,\\
&u^{\prime}(1) + \sqrt{\lambda}\gamma u(1) = 0,\\
\end{split}
\end{equation*}
where $\lambda$, $c$, $K$, and $\gamma$ are positive parameters and the parameter $L\in(0,\frac{1}{2})$. The steady state reaction-diffusion equation above occurs in ecological systems and population dynamics. The above model exhibits logistic growth in the one-dimensional habitat $\Omega_0=(0,1)$, where grazing (type of predation) is occurring on the subregion $[L,1-L]$. In this model, $u$ is the population density and $c$ is the maximum grazing rate. $\lambda$ is a parameter which influences the equation as well as the boundary conditions, and $\gamma$ represents the hostility factor of the surrounding matrix. Previous studies have shown the occurrence of S-shaped bifurcation curves for positive solutions for certain parameter ranges when the boundary condition is Dirichlet ($\gamma \longrightarrow \infty$). Here we discuss the occurrence of S-shaped bifurcation curves for certain parameter ranges, when $\gamma$ is finite, and their evolutions as $\gamma$ and $L$ vary.
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